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The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics

Part of the The Western Ontario Series in Philosophy of Science Book Series (volume 82)

DOI: 10.1007/978-3-030-18707-1

ISBN: 978-3-030-18706-4

Publisher: Springer International Publishing - Springer Journals

Sep 10, 2019

1 - 10 of 10 Chapters

LitStream

The Continuous and the Discrete in Ancient Greece, the Orient, and the European Middle Ages

Bell, John L.

Pages: 3 –39

[The opposition between the Continuous and the Discrete played a significant role in ancient Greek philosophy. This probably derived from the still more fundamental opposition concerning the One and the Many, an antithesis lying at the heart of early Greek thought.]
LitStream

The Sixteenth and Seventeenth Centuries. The Founding of the Infinitesimal Calculus

Bell, John L.

Pages: 41 –76

[The early modern period saw the spread of knowledge in Europe of ancient geometry, particularly that of Archimedes, and a loosening of the Aristotelian grip on thinking. In regard to the problem of the continuum, the focus shifted away from metaphysics to technique, from the problem of “what...
LitStream

The Eighteenth and Early Nineteenth Centuries: The Age of Continuity

Bell, John L.

Pages: 77 –104

[The leading practitioner of the calculus, indeed the leading mathematician of the eighteenth century, was Leonhard Euler (1707–83). While Euler’s genius has been described as being of “equal strength in both of the main currents of mathematics, the continuous and the discrete”, philosophically...
LitStream

The Reduction of the Continuous to the Discrete in the Nineteenth and Early Twentieth Centuries

Bell, John L.

Pages: 105 –146

[The rapid development of mathematical analysis in the eighteenth century had not concealed the fact that its underlying concepts not only lacked rigorous definition but were even (e.g. in the case of differentials and infinitesimals) of doubtful logical character. The lack of precision in the...
LitStream

Dissenting Voices: Divergent Conceptions of the Continuum in the Nineteenth and Early Twentieth Centuries

Bell, John L.

Pages: 147 –184

[Despite the great success of Weierstrass, Dedekind and Cantor in constructing the continuum from arithmetical materials, a number of thinkers of the late nineteenth and early twentieth centuries remained opposed, in varying degrees, to the idea of explicating the continuum concept entirely in...
LitStream

Topology

Bell, John L.

Pages: 187 –195

[In the late nineteenth and early twentieth century the investigation of continuity led to the creation of topology, a major new branch of mathematics conferring on the idea of the continuous a vast generality. The origins of topology lie both in Cantor’s theory of sets of points as well as the...
LitStream

Category/Topos Theory

Bell, John L.

Pages: 197 –207

[Category theory is a framework for the investigation of mathematical form and structure in their most general manifestations. Central to it is the concept of structure-preserving map, or transformation. While the importance of this notion was long recognized in geometry (witness, for example,...
LitStream

Nonstandard Analysis

Bell, John L.

Pages: 209 –213

[Once the continuum had been provided with a set-theoretic foundation, the use of the infinitesimal in mathematical analysis was largely abandoned. And so the situation remained for a number of years. The first signs of a revival of the infinitesimal approach to analysis surfaced in 1958 with a...
LitStream

The Continuum in Constructive and Intuitionistic Mathematics

Bell, John L.

Pages: 215 –231

[In constructive mathematics, a problem is counted as solved only if an explicit solution can, in principle at least, be produced. Thus, for example, “There is an x such that P(x)” means that, in principle at least, we can explicitly produce an x such that P(x). If the solution to the problem...
LitStream

Smooth Infinitesimal Analysis/Synthetic Differential Geometry

Bell, John L.

Pages: 233 –271

[Mathematicians have developed two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the (much older) idea behind the synthetic approach is to...

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